Question

How do you integrate \[\dfrac{x}{x+10}dx\]?

Answer 1

Hint: This question belongs to the integration chapter. So, we should have a better knowledge of integration. While solving the question, we will first write the equation \[\dfrac{x}{x+10}\] in a simple form, so that we can find the integration easily. After that, we arrange them and find the integration.

Complete step by step answer:
Let us solve the question.
We have to integrate \[\dfrac{x}{x+10}dx\].
Before solving this question, let us recall some formulas which are going to be used in this question for solving. These formulas are used to make the questions solving process easy. The formulas are:\[\int{1dx}=x+{{C}_{1}}\] and \[\int{\dfrac{1}{x}dx}=\ln x+{{C}_{2}}\] where \[{{C}_{1}}\] and \[{{C}_{2}}\] are some contants.
Let \[I=\int{\dfrac{x}{x+10}dx}\]
So, we write the above equation in simple form to make the integration easier.
\[I=\int{\dfrac{x}{x+10}dx}=\int{\left( \dfrac{x+10-10}{x+10} \right)dx}\]
The above equation also can be written as
\[\Rightarrow I=\int{\left( 1-\dfrac{10}{x+10} \right)dx}\]
Which is also can be written as
(using the sum rule for integration)
\[\Rightarrow I=\int{1dx}-\int{\dfrac{10}{x+10}dx}\]
As we know that integration of 1 is x. Therefore, using the formula: \[\int{1dx}=x+{{C}_{1}}\] , where \[{{C}_{1}}\] is any constant.
Hence, we can write
\[\Rightarrow I=x+{{C}_{1}}-\int{\dfrac{10}{x+10}dx}\]
Now, we just have to integrate \[\dfrac{10}{x+10}\] with respect to x.
As we know that integration of \[\dfrac{1}{x}\] is\[\ln x\]. Therefore, using the formula: \[\int{\dfrac{1}{x}dx}=\ln x+{{C}_{2}}\] where \[{{C}_{2}}\] is any constant.
So, we can say that \[\int{\dfrac{1}{x+10}dx}=\ln \left( x+10 \right)+{{C}_{2}}\]
\[\int{\dfrac{10}{x+10}dx}=10\ln \left( x+10 \right)+{{C}_{2}}\]
Hence,
\[I=x+{{C}_{1}}-\int{\dfrac{10}{x+10}dx}=x+{{C}_{1}}-10\ln \left( x+10 \right)-{{C}_{2}}\]
As \[{{C}_{1}}-{{C}_{2}}\] is also a constant. Let us name it as C.
\[I=x-10\ln \left( x+10 \right)+C\]
So, above is the integration of\[\dfrac{x}{x+10}dx\].

Note:
 For solving this type of problems, remember the formulas of integration. For example, we have used some formulas here. The formulas are:\[\int{1dx}=x+{{C}_{1}}\] and \[\int{\dfrac{1}{x}dx}=\ln x+{{C}_{2}}\].
We can solve this question using a partial fraction method.
We can write \[\dfrac{x}{x+10}\] as
\[\dfrac{x}{x+10}=A+\dfrac{B}{x+10}\]
\[\Rightarrow x=\left( x+10 \right)A+\dfrac{B}{x+10}\times \left( x+10 \right)=Ax+10A+B\]
\[\Rightarrow x=Ax+10A+B\]
From the above equation, we can say that x=1 and B=-10
Therefore, \[\dfrac{x}{x+10}=1+\dfrac{-10}{x+10}=1-\dfrac{10}{x+10}\]
Hence, \[\int{\dfrac{x}{x+10}dx}=\int{\left( 1-\dfrac{10}{x+10} \right)dx}\]
Now, from here we have calculated above in the question.
So, \[\int{\left( 1-\dfrac{10}{x+10} \right)dx}=x-10\ln \left( x+10 \right)+C\]
From this method also, we get that
\[\int{\dfrac{x}{x+10}dx}=x-10\ln \left( x+10 \right)+C\]
Hence, from here also we get the same value. So, we can use this method also.